Covering Spaces

Abstract

The concept of a covering space is a valuable source of examples, applications, and problems, as well as a basis for new ideas. Our analysis begins with an exploration of the lifting problem for a map into the base space. When the mapping is restricted to be a closed loop, the resulting structure is seen to be the fundamental group, and this provides a framework within which the lifting properties may be recast in algebraic terms. Continuing with this connection, the relations among the covering spaces over a given base are expressed in terms of the subgroups of the fundamental group of the base. The chapter closes with an examination of the relationship between the fundamental group and the first homology group and a discussion of Van Kampen’s Theorem, a useful computational tool.